Riemann’s Rearrangement Theorem

Themistoklis Haris

This post will be devoted to a very special and surprising result given by the great Bernhard Riemann on the topic of the convergence of infinite series.

The theorem states:

Let \{a_n\}_{n=1}^{\infty} be a sequence of real numbers so that \sum\limits_{n=1}^{\infty} a_n is conditionally convergent (it converges but not absolutely). Then if we let M\in\mathbb{R}\cup\{\infty,-\infty\}, there exists a permutation \sigma of \{a_n\} such that \sum\limits_{n=1}^{\infty} a_{\sigma(n)} = M

Let’s let that result sink in first through a classic example. Consider the series 1 - \frac{1}{2} + \frac{1}{3} - \cdots = \sum\limits_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}. We know this series converges to ln(2), through Taylor’s series but the series \sum\limits_{n=1}^{\infty}\frac{1}{n} is the harmonic series and diverges. So our series is conditionally convergent. But observe that we can rearrange the terms of the infinite sum so that the series converges to the half of itself:

\left(1-\frac{1}{2}-\frac{1}{4}\right) + \left(\frac{1}{3}-\frac{1}{6} - \frac{1}{8}\right) + \left(\frac{1}{5} - \frac{1}{10} - \frac{1}{12}\right) + \cdots = \sum\limits_{n=1}^{\infty}\left(\frac{1}{2n-1}-\frac{1}{4n-2} - \frac{1}{4n}\right) = \sum\limits_{n=1}^{\infty}\frac{1}{4k(2k-1)} = \frac{1}{2}ln(2)

 

This is very weird. The same numbers added up in different order up to infinity, lead to a different sum. So addition works in strange ways when we take limits to infinity. We already sort of knew that but this theorem showcases it to an extent I couldn’t imagine before. And not only can we get half our sum by rearranging the terms, but we can get all possible numbers, by rearranging the terms in some way. We can even get \pm \infty.

We should prove this important theorem. In fact we shall prove a stronger statement: that if -\infty \leq a \leq b \leq \infty, there exists a rearrangement \sigma of \{a\}_{n=1}^{\infty} such that

\liminf_{n\rightarrow\infty}s_n'= a and \limsup_{n\rightarrow \infty}s_n' = b,

where s_n' is the sequence of partial sums of the rearranged sequence.

We get our result immediately if a=b. In case there is confusion about the \limsup and \liminf notation, recall that these signify the limit superior and limit inferior of a sequence, which are limiting bounds on a sequence (more info here).

First, define the following sequences: p_n = \frac{|a_n| + a_n}{2} and q_n = \frac{|a_n|-a_n}{2}. So p consists of all the positive terms in a and of 0s in the place of the negative terms and q is the opposite.

FACT: p and q diverge.

This isn’t very hard to see. If they both converged then \sum|a_n| would have to converge and if one converged and one diverged then \sum a_n would converge (check this!)

Now consider the sequences P_n, Q_n of positive terms and of the absolute values of the negative terms in order in \sum a_n. P and Q differ from p and q only in the placement of 0s in between of the latters’ terms, so both P and Q diverge.

IDEA / GOAL: Construct sequences \{m_n\}, \{k_n\} such that the series

P_1 + \cdots P_{m_1} - Q_1 -\cdots - Q_{k_1} + \\[1mm] P_{m_1+1} + \cdots + P_{m_2} - Q_{k_1+1} - \cdots - Q_{k_2} + \cdots

,which is obviously a rearrangement of \sum a_n satisfy our theorem’s target.

How to do this? First choose sequences \alpha_n \rightarrow a, \beta_n \rightarrow b so that \beta_1 > 0 and \alpha_n < \beta_n. Then let m_n and k_n be the smallest integers such that

x_n = \sum\limits_{i=0}^{m_n}P_i - \sum\limits_{i=0}^{k_{n-1}} Q_i > \beta_n and y_n = \sum\limits_{i=0}^{m_n}P_i - \sum\limits_{i=0}^{k_n} Q_i < \alpha_n (let k_0 = -1)

We can always find such integers because P, Q diverge.

Since m_n and k_n are the smallest integers with this property, we have that |x_n - \beta_n| \leq P_{m_n} and |y_n - \alpha_n| \leq Q_{k_n} so that x_n \rightarrow a and y_n \rightarrow b.

Finally, it is clear that no subsequential limit of our rearranged series can converge to something larger than b or smaller than a and so our proof is complete.

Proof adapted from the book “Principles of Mathematical Analysis”, by Walter Rudin.